LFVand(g-2)innon-universalSUSYmodelswithlighthiggsinos · LL = M2 L + v2 d 2 Yy Y ‘ + 1 m 2 Z...
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IFIC/20-10, FTUV-20-0313 LFV and (g-2) in non-universal SUSY models with light higgsinos C. Han 11 , M.L. López-Ibáñez † 2 , A. Melis ‡ 3 , O. Vives ‡ 4 , L. Wu ?5 , J.M. Yang †, § 6 * School of Physics, KIAS, 85 Hoegiro, Seoul 02455, Republic of Korea. † CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics Chinese Academy of Sciences, Beijing 100190, P. R. China. ‡ Departament de Física Tèorica, Universitat de València & IFIC, Universitat de València & CSIC, Dr. Moliner 50, E-46100 Burjassot (València), Spain. ? Department of Physics and Institute of Theoretical Physics, Nanjing Normal University Nanjing 210023, P. R. China. § School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China Abstract We consider a supersymmetric type-I seesaw framework with non-universal scalar masses at the GUT scale to explain the long-standing discrepancy of the anomalous magnetic moment of the muon. We find that it is difficult to accommodate the muon g-2 while keeping charged-lepton flavor violating processes under control for the conventional SO(10)-based relation between the up sector and neutrino sector. However, such tension can be relaxed by adding a Georgi-Jarlskog factor for the Yukawa matrices, which requires a non-trivial GUT-based model. In this model, we find that both observables are compatible for small mixings, CKM-like, in the neutrino Dirac Yukawa matrix. 1 Introduction After the discovery of neutrino oscillations in 1998 by Superkamiokande [1], the Standard Model (SM) was forced to include massive neutrinos. Yet, the smallness of their masses seems to require a new framework different from the SM Yukawa couplings in the charged lepton or quark sectors. Simultaneously, it was confirmed experimentally that neutrino mass eigenstates are a non-trivial combination of the flavour states. Consequently, we know that the family lepton numbers, L e ,L μ and L τ , are violated in Nature. This necessarily implies some degree of violation in the charged- lepton sector, although it has not yet been observed. Several mechanisms have been concocted to explain the extreme smallness of neutrino masses and all of them require the existence of new physics (NP). Unfortunately, the available experimental information on the mass splittings and mixing is still insufficient to disclose the physics behind 1 [email protected] 2 [email protected] 3 [email protected] 4 [email protected] 5 [email protected] 6 [email protected] 1 arXiv:2003.06187v1 [hep-ph] 13 Mar 2020
Transcript of LFVand(g-2)innon-universalSUSYmodelswithlighthiggsinos · LL = M2 L + v2 d 2 Yy Y ‘ + 1 m 2 Z...
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IFIC/20-10, FTUV-20-0313
LFV and (g-2) in non-universal SUSY models with light higgsinos
C. Han11 , M.L. López-Ibáñez†2 , A. Melis‡3 , O. Vives‡4 , L. Wu?5 , J.M. Yang†, §6
∗ School of Physics, KIAS, 85 Hoegiro, Seoul 02455, Republic of Korea.† CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics
Chinese Academy of Sciences, Beijing 100190, P. R. China.‡ Departament de Física Tèorica, Universitat de València & IFIC, Universitat de València & CSIC,
Dr. Moliner 50, E-46100 Burjassot (València), Spain.? Department of Physics and Institute of Theoretical Physics, Nanjing Normal University
Nanjing 210023, P. R. China.§ School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, P. R. China
Abstract
We consider a supersymmetric type-I seesaw framework with non-universal scalar masses atthe GUT scale to explain the long-standing discrepancy of the anomalous magnetic momentof the muon. We find that it is difficult to accommodate the muon g-2 while keepingcharged-lepton flavor violating processes under control for the conventional SO(10)-basedrelation between the up sector and neutrino sector. However, such tension can be relaxedby adding a Georgi-Jarlskog factor for the Yukawa matrices, which requires a non-trivialGUT-based model. In this model, we find that both observables are compatible for smallmixings, CKM-like, in the neutrino Dirac Yukawa matrix.
1 Introduction
After the discovery of neutrino oscillations in 1998 by Superkamiokande [1], the Standard Model(SM) was forced to include massive neutrinos. Yet, the smallness of their masses seems to requirea new framework different from the SM Yukawa couplings in the charged lepton or quark sectors.Simultaneously, it was confirmed experimentally that neutrino mass eigenstates are a non-trivialcombination of the flavour states. Consequently, we know that the family lepton numbers, Le, Lµand Lτ , are violated in Nature. This necessarily implies some degree of violation in the charged-lepton sector, although it has not yet been observed.
Several mechanisms have been concocted to explain the extreme smallness of neutrino masses andall of them require the existence of new physics (NP). Unfortunately, the available experimentalinformation on the mass splittings and mixing is still insufficient to disclose the physics behind
[email protected]@[email protected]@[email protected]@itp.ac.cn
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their true origin. New observables are required to improve our understanding and charged-lepton-flavour violation (CLFV) is one of the best options at our reach. Nevertheless, theabsence of any signal of CLFV may indicate that the associated NP is considerably heavy.On the other hand, the persistent discrepancy between the experimental measurement of theanomalous magnetic moment of the muon [2] and its SM prediction [3],
∆aµ = aexpµ − aSMµ = (2.7± 0.7)× 10−9, (1)
remains an interesting motivation to explore new models [4].
With respect to neutrino masses, the type-I seesaw mechanism [5] seems to be the most naturalextension of the SM to generate them. It assumes the existence of right-handed neutrinos (RHν),which are singlets of the SM and, therefore, are allowed to have both Dirac and Majorana massterms by the gauge symmetries. Mediated by them, an effective dimension-5 Weinberg operator[6] would be induced, producing the light neutrino masses after the electroweak (EW) symmetrybreaking. Although RHν may live at any scale between the EW and the GUT scale, a naïvedimensional analysis of the Weinberg operator signals a Majorana mass around the usual GUTscale ∼ 1015−16 GeV. That means that the SM supplemented with RHν at high energies suffersfrom a serious hierarchy problem [7].
Despite the fact that no signal of superparticles has been found at LHC, supersymmetric modelsmay still be the appropriate candidate to alleviate this problem and may also answer otherquestions, such as the nature of dark matter [8] and the exact unification of the gauge couplingsat the GUT scale [9]. In addition, the presence of right-handed neutrinos induces slepton mixingthrough the renormalization group equation (RGE), which may produce visible CLFV effects[10, 11]. In particular, SUSY models with non-universal scalar masses at the GUT scale seem tobe favoured from naturalness considerations [12], allowing for a Higgs boson mass at 125 GeVand low electroweak fine-tuning [13]. Here we will analyse supersymmetric non-universal Higgsmodels with an additional parameter for the third generation of scalar superpartners (NUHM3)[14]. We focus on the so-called light higgsino-world scenario [15–18] in which the SUSY matterscalars are pushed into the multi-TeV scale while µ . 1 TeV, as natural SUSY requires.
The paper is organised as follows: in Section 2, the main ingredients of a supersymmetric type-Iseesaw model are presented. We also comment on some specific features common to GUT-motivated scenarios. In Section 3, the effect of the right-handed neutrinos on the running of theslepton soft masses is described. In Section 4, we discuss the main observables of our analysisproviding some useful analytic relations between CLFV processes and (g − 2)µ. Section 5 isdevoted to the results of our numerical scan. We summarize our conclusions in Section 6.
2 Supersymmetric type-I seesaw
The supersymmetric type-I seesaw considers the MSSM particle content augmented with threechiral superfields, one for each generation of right-handed neutrinos (RHν). The interactionsamong chiral supermultiplets are determined by the superpotential which, in this framework,contains new interactions involving RHν:
W = WMSSM + νc TR Yν `L ·Hu +
1
2νc TR MR ν
cR , (2)
with
WMSSM = ec TR Y` `L ·Hd + dc TR YdQL ·Hd (3)
+ uc TR YuQL ·Hu + µHd ·Hu. (4)
2
-
νL ν†R ν
†R νL
H0u H0u
Figure 1.- Feynman diagram associated with the type-I seesaw mechanism. The dimension-5 Weinbergoperator is effectively generated when the heavy right-handed neutrinos are integrating out.
The SUSY soft-breaking interactions introduce mass terms for the superpartners as well astrilinear couplings between the new sfermions and the Higgs. The relevant terms in our case,not including the quark sector, are:
−Lsoft =1
2
(M1B̃B̃ + M2W̃W̃ + M3g̃g̃ + c.c.
)(5)
+ ˜̀†LM2˜̀
˜̀L + ẽ
c TR M
2ẽ ẽ
c ∗R + ν̃
c TR M
2ν̃ ν̃
c ∗R (6)
+(ẽc †R Ae
˜̀L ·Hd + ν̃c †R Aν ˜̀L ·Hu + c.c
)(7)
+ m2Hd H∗uHu + m
2HdH∗dHd + . . . (8)
At an energy scale above the heavy RHν mass, µ ≥ mνc3 , an effective operator with the left-handed neutrinos and Higgs fields is generated by the process in Fig. 1. Integrating out theheavy neutrinos and replacing the Higgs by its vev at low energies, the following mass matrix isproduced:
Mν = −υ2u2Y Tν M
−1R Yν , (9)
where υu = υh sinβ and υh = 246 GeV. Neutrino oscillations are directly related to Mν inEq. (9) and provide information about the squared-mass differences and mixing of neutrinos, seeTable 1. However, those measurements are not enough to fully reconstruct the UV parametersof the model, namely Yν and MR.
Extensions of the SMmay provide additional information through related observables, like CLFVprocesses. One of the best examples is supersymmetric extensions of the SM supplemented witha type-I seesaw, where slepton soft-breaking masses are related to the neutrino Yukawa couplings.However, additional assumptions are usually made to simplify the analysis of the phenomenologyof these models. The minimal strategy consists in presuming universal soft-masses and a concretestructure for the neutrino Yukawa couplings at the GUT scale. Then non-universal entries aregenerated in the soft masses through the renormalization group evolution (RGE), proportional tothe hypothesised Yukawa couplings. It is important to remark that these RGE contributions arealways present in SUSY models irrespective of the presence of additional non-universal entriesat the GUT scale. Therefore, barring accidental cancellations, these effects are the minimaloutcome of supersymmetric seesaw models.
In this framework, we perform the analysis in two limit cases: one in which the rotation anglesin V νL are small, or CKM-like, and another where the mixing is large, or PMNS-like. Thesetwo scenarios should span any intermediate possibility so that general conclusions can be fairlyderived. For instance, based on an underlying SO(10) gauge symmetry, one may consider thatYν and Yu, in the basis of diagonal Yukawas for charged leptons and down-type quarks, are
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Observable Normal Hierarchy Inverted Hierarchy
θ12 (o) 33.82+0.78−0.76 33.82
+0.78−0.76
θ23 (o) 48.3+1.1−1.9 48.6
+1.1−1.5
θ13 (o) 8.61+0.13−0.13 8.65
+0.13−0.12
∆m212 (10−5 eV2) 7.39+0.21−0.20 7.39
+0.21−0.20
∆m23` (10−3 eV2) 2.523+0.032−0.030 −2.509+0.032−0.030
Table 1 .- Global fit on neutrino observables by NuFIT 4.1. (2019) [19]. Similarresults have been found in [20].
deeply connected at the unification scale through relations such as [21–24]:
- Small Mixing (CKM-like): Y ckmν = kGJ Yu. (10)
- Large Mixing (PMNS-like): Y pmnsν = kGJ Ydiagu V
Tpmns (11)
where, in these equations, we have introduced a Georgi-Jarlskog (GJ) factor, kGJ, which mayarise in some GUT realisations due to the different representations of the unified group that mixto generate the SM Higgs doublet.
Within a SO(10) model, for example, if the dominant contribution to the Yukawa interactionsis due to a Higgs field transforming as a 10, a perfect unification between neutrinos (chargedleptons) and up-type (down-type) quarks is expected, so kGJ = 1. Conversely, if the dominantcontribution comes from a 126 representation, a factor kGJ = 3 appears between leptons andquarks. Another possibility is having an effective Higgs field transforming as a 120, which maybe the product of a 45 and 10 representations. The 45 can acquire a nonzero vev in the flatdirection B−L+κT3, which preserves the SM and distinguishes between RH fermions7 [25]. Infact, complete flavour models usually require the combined effect of more than one representationto generate dissimilar hierarchies among generations [26]. For instance, unification in the downsector as in the classical Georgi-Jarslkog scheme demands |yτ/yb|GUT = 1 and |yµ/ys|GUT =kGJ = 3. However, nowadays, these relations are no longer favoured phenomenologically [27–30]but the updated range ∣∣yµ/ys
∣∣GUT
= [ 2.5, 6.5 ], (12)
obtained in [29]. This is the reason we take to generalize our kGJ factor to kGJ = B−L+κT3. Inthe case of neutrino Yukawa couplings, we are allowed to consider that the dominant contributionto the up and neutrino Yukawas comes from the representation giving rise to this kGJ factor.Then, varying κ in the interval where Eq. (12) is satisfied, one observes that
∣∣Yν∣∣ = [ 0, 1/2 ]
∣∣Yu∣∣. (13)
This is the interval taken for the kGJ factor in our numerical analysis.
We explore the parameter space of seesaw NUHM3 models in which some of the stringentconditions of the typical mSUGRA models are relaxed. We introduce three additional degreesof freedom in the scalar soft-breaking sector: instead of one common scalar mass, we will considerthe following four
m(1,2)0 6= m
(3)0 6= mHu 6= mHd (14)
7T3 refers to the third component of a SU(2)R gauge group which is spontaneously broken afterwards.
4
http://www.nu-fit.org
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where we have included a non-universal, but diagonal, charged-slepton mass matrix in the basisof diagonal charged-lepton Yukawa couplings at the GUT scale. The supersymmetric sector ofthe models is determined by five parameters at the GUT scale,
m(1,2)0 , m
(3)0 , M1/2, tanβ, A0, (15)
and two more at the EW scale,µ, MA0 , (16)
which can be taken in exchange of mHu and mHd ,
3 RGEs and lepton flavour violation
The introduction of RH neutrinos makes the effect of the RGEs specially relevant for sleptons[10, 11]. During the running, the heavy neutrinos induce off-diagonal entries in the sleptonsoft terms through radiative corrections. Those flavour-violating interactions allow for CLFVprocesses that otherwise, within the SM, would be greatly suppressed by the neutrino masses.The main effect occurs for the soft-mass matrices of the LH sleptons and can be worked out bysolving the RGEs
µd
dµ
(M2˜̀
L
)= µ
d
dµ
(M2˜̀
L
)
MSSM
(17)
+1
16π2
(M2˜̀
LY †ν Yν + Y
†ν YνM
2˜̀L
+ 2(Y †νM
2ν̃Yν + m
2HuY
†ν Yν + A
†νAν
)), (18)
where the first term denotes the MSSM contribution in the absence of RHν,
µd
dµ
(M2˜̀
L
)
MSSM
=1
16π2
(M2˜̀
LY †` Y` + Y
†` Y`M
2˜̀L
+ 2(Y †` M
2ẽ Y` + m
2HdY †` Y` + A
†`A`
))(19)
− 1(
6
5g21 |M1|2 + 6 g22 |M2|2
)+ 1
3
5g21S (20)
with S ≡ Tr[M2Q̃L
+ M2d̃− 2M2ũ −M2˜̀
L+ M2ẽ ] − m2Hd + m
2Hu
. In the basis of diagonal RHνand charged-lepton Yukawas, the leading log approximation is proportional to the square of theneutrino Yukawas as:
(M2˜̀
)i 6=j
' −2m20 +m
2Hu
+A2016π2
∑
k
Y ∗ν,kiYν,kj log
(m2GUTm2Nk
), (21)
where we take the limit m(1,2)0 ' m(3)0 ' m0 and approximate mGUT to be of the order of the
scale at which the soft terms appear in the Lagrangian (the typical scale of SUSY-breakingtransmission). The main effect, with hierarchical Yukawas, is due to the heaviest Majorananeutrino and happens before its decoupling at µ > mνc3 . Trilinear couplings receive similarcorrections, although they have a smaller impact on the CLFV observables studied here. Incontrast, no flavour violation is produced in the RH charged-slepton sector at one-loop, sincethe RGEs only depend on Y` and on the gauge couplings, hence they are diagonal in the basiswhere Y` is diagonal. The off-diagonal elements produced radiatively enter the total 6×6 sleptonmass matrix as small insertions (compared to the diagonal terms) in the LL and LR/RL sector:
M2˜̀ =
(∆LL ∆LR∆†LR ∆RR
)(22)
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∆LL = M2˜̀L
+v2d2Y †` Y` + 1 m
2Z cos 2β
(−1
2+ sin2 θw
)(23)
∆RR = M2ẽR
+v2d2Y †` Y` − 1 m2Z cos 2β sin2 θw (24)
∆LR =vd√
2(A` − µ∗ Y` tanβ) . (25)
4 Observables
Up to now, we have defined the supersymmetric model that we analyse in this project andits RGE evolution to the electroweak scale. The next step will be to compare its predictionswith the low-energy observables, to constrain the allowed parameter space or to find possiblediscrepancies from the SM predictions.
The first observable we have to reproduce is the recently measured value of the Higgs mass,which is a strong constraint on any supersymmetric extension of the SM. Then, as we aremainly interested in the leptonic sector, we concentrate on two main observables: the anomalousmagnetic moment of the muon and the CLFV process µ→ eγ.
4.1 Higgs mass
Previous works have extensively discussed how to accommodate the 125 GeV observed Higgsboson [31] within a minimal supersymmetric framework [32–35]. In the MSSM, it is known thatthe tree-level value of the lightest Higgs mass is bounded from above by MZ whilst radiativecorrections, coming from the fermion-sfermion loops, may increase it up to 135 GeV [36]. Asthose corrections are proportional to the corresponding fermion Yukawa couplings, the dominantcontribution is due to the top-stop diagram and can be written as
∆mh '3
4π2cos2 α y2t m
2t
[ln
(mt̃1mt̃2m2t
)+ ∆thr
], (26)
where α is the mixing angle between the scalar components of H0u and H0d after EWSB and∆thr stands for the threshold corrections dependent on the stop mixig [37]. While constrainedversions of the MSSM, such as mSUGRA, GMSB or AMSB, usually have difficulties to generatethe observed mass, scenarios with non-universal conditions at the GUT scale are able to improvetheir predictions and provide realisations with a low amount of fine-tuning [13, 33].
In the models considered here, the stop mass is determined by the scalar mass parameter m(3)0at the GUT scale. As we will see below, in order to obtain the adequate mass, large values form
(3)0 are expected,
m(3)0 & 4 TeV, (27)
with stops masses in the few-TeV regime,
mt̃1 , mt̃2 & 2.5 TeV. (28)
4.2 Anomalous magnetic moment of the muon
In the MSSM, leptons receive supersymmetric corrections to their anomalous magnetic momentdue to neutralino and chargino loops that effectively generate the dipole operators, defined in
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µR ν̃Lµ µL
H̃−d
H̃+u W̃−
W̃+
v sinβµ M2
γ γ
µR ν̃Lµ eLν̃Le
M2
ℓ̃
21
H̃−d
H̃+u W̃−
W̃+
v sinβµ M2
Figure 2.- Diagrammatic representation of the leading contribution to asusyµ (left) and BR(µ → eγ)(right) in our models.
Eq. (58) in Appendix B.2, where we can find the full expressions, [11, 38],
asusy` = a(c)` + a
(n)` . (29)
The supersymmetric amplitude is usually dominated by the processes where the chirality flip ofthe fermion occurs at the vertex, which is proportional to the Yukawa coupling and thereforetanβ-enhanced. The mass insertion approximation (MIA) [39–42] allows us to see this explicitlyby means of expanding the full amplitude, extracting the relevant diagrams and identifying themain parameters. It has been implemented in Appendix B.2. The diagramatic interpretation ofthe dominant processes is depicted in Fig. 2 (left).
In the light higgsino-world scenario, the LSP is the neutralino, which is mainly higgsino andquasi-degenerate in mass with the second-lightest neutralino (NLSP) and the lightest chargino.We observe that the region where asusyµ is within the 3σ range exhibits the following hierarchiesbetween masses: µ�M2 . m˜̀
L. Then, the process is expected to be dominated by the chargino
loop8, since its loop function for x = µ2/m2˜̀L� 1 is the largest one (see Fig. 9). Therefore,
asusyµ ' −α24π
m2µm2ν̃µ
M2 µ
M22 − µ2F c2 (x2ν̃µ , xµν̃µ) tanβ, (30)
where x2ν̃µ = M22 /m2ν̃µ , xµν̃µ = µ2/m2ν̃µ and F
c2 (x1, x2) ≡ f
(c)2 (x1)− f
(c)2 (x2) with f
c2(x) the loop
function provided in Appendix A. We compare the exact result worked out by SPheno-4.0.4[43, 44] versus the MIA expression in Fig. 3 (left) and notice that the second works quite wellfor most of the points. Some deviations appear for isolated points in the region where asusyµ isvery small and contributions from other diagrams may compete and become important.
From Eq. (30), another phenomenological consequence can be inferred: a SUSY contributionthat accounts for the current discrepancy between the experimental and the SM theoretical valueof the muon anomalous magnetic moment will require light sneutrinos in the second generation.In the models analysed here, the masses for sfermions of the first two generations are determinedby the scalar massm(1,2)0 at the GUT scale. Therefore, contrary tom
(3)0 (see discussion in Section
4.1), we expect quite small values for m(1,2)0 to reproduce ∆aµ,
m(1,2)0 � m
(3)0 . (31)
In practice, Eq. (31) leads to a decoupled spectrum for sfermions where the third generation issignificantly heavier than the first two ones.
8More details about the derivation of the chargino and neutralino dominant terms under the MIA can befound in Appendix B.2.
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�
